Let $\Omega = \cup_{i = 1}^{N} K_i$ be a bounded domain, $P_h$ be the k-th piecewise polynomial space with some restrictions such that for all $v_h\in P_h$, we can define $$ \|v_h\|_h := \sum_{i }^N \int_{K_i} |\nabla v_h|^2 $$ as the norm on $P_h$.
By the definition, we know that $\|\cdot\|_h$ also can be the norm in $H_0^1(\Omega)$. So can $\|\cdot\|_h$ be the norm on $H_0^1 + P_h$?
Actually firstly I am wondering whether I should consider $H_0^1, P_h$ as the subspace in $L^2$ or other spaces?
This is not a norm on $P_h$, since any nonzero piecewise constant function on the triangulation has norm equal to 0 (and so the homogeneity property required for a norm is not satisfied). You would need to for instance add the square integral of the jumps over internal edges of the triangulation, plus the integral over some subset of the boundary of positive boundary measure.
However, this norm is a norm on $H^1_0$, and therefore would be a norm on the intersection $H^1_0\cap P_h$.